Learn about this topic in these articles:

complex analysis

...name). These integrals cannot be evaluated explicitly; they do not define a function that can be obtained from the rational and trigonometric functions, a difficulty that added to their interest. Elliptic integrals were intensively studied for many years by the French mathematician Adrien-Marie Legendre, who was able to calculate tables of values for such expressions as functions of their...

work of

Landen

Legendre

In 1786 Legendre took up research on
elliptic integrals. In his most important work,
Traité des fonctions elliptiques (1825–37; “Treatise on Elliptic Functions”), he reduced
elliptic integrals to three standard forms now known by his name. He also compiled tables of the values of his
elliptic integrals and showed how they can be used to solve important...